LMFDB - Embedded newform 1024.4.b.k.513.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,4,Mod(513,1024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1024.513");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$60.4179558459$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 36x^{8} + 405x^{6} + 1380x^{4} + 420x^{2} + 32 $ x^10 + 36x^8 + 405x^6 + 1380x^4 + 420x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			513.10
Root			$-2.34476i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.4.b.k.513.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.43597i q^{3} +12.2748i q^{5} +1.63924 q^{7} -44.1656 q^{9} +O(q^{10})$	$q+8.43597i q^{3} +12.2748i q^{5} +1.63924 q^{7} -44.1656 q^{9} +25.7416i q^{11} +13.2187i q^{13} -103.550 q^{15} -53.6113 q^{17} +100.391i q^{19} +13.8286i q^{21} +25.1189 q^{23} -25.6706 q^{25} -144.809i q^{27} -256.105i q^{29} -132.684 q^{31} -217.155 q^{33} +20.1214i q^{35} +247.306i q^{37} -111.512 q^{39} -198.660 q^{41} -404.064i q^{43} -542.124i q^{45} +78.3629 q^{47} -340.313 q^{49} -452.264i q^{51} +743.559i q^{53} -315.973 q^{55} -846.894 q^{57} -65.8036i q^{59} +273.392i q^{61} -72.3982 q^{63} -162.256 q^{65} -399.066i q^{67} +211.903i q^{69} +727.536 q^{71} -106.065 q^{73} -216.556i q^{75} +42.1968i q^{77} -58.9970 q^{79} +29.1298 q^{81} -580.049i q^{83} -658.068i q^{85} +2160.50 q^{87} +768.959 q^{89} +21.6686i q^{91} -1119.32i q^{93} -1232.28 q^{95} -809.953 q^{97} -1136.89i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 28 q^{7} - 54 q^{9}+O(q^{10}) $ 10 * q + 28 * q^7 - 54 * q^9	$ 10 q + 28 q^{7} - 54 q^{9} - 124 q^{15} + 4 q^{17} + 276 q^{23} - 50 q^{25} - 368 q^{31} - 4 q^{33} + 732 q^{39} - 944 q^{47} - 94 q^{49} + 1380 q^{55} - 108 q^{57} - 2628 q^{63} - 492 q^{65} + 3468 q^{71} + 296 q^{73} - 4416 q^{79} - 482 q^{81} + 6036 q^{87} - 88 q^{89} - 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 28 * q^7 - 54 * q^9 - 124 * q^15 + 4 * q^17 + 276 * q^23 - 50 * q^25 - 368 * q^31 - 4 * q^33 + 732 * q^39 - 944 * q^47 - 94 * q^49 + 1380 * q^55 - 108 * q^57 - 2628 * q^63 - 492 * q^65 + 3468 * q^71 + 296 * q^73 - 4416 * q^79 - 482 * q^81 + 6036 * q^87 - 88 * q^89 - 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	8.43597i	1.62350i	0.584003	+	0.811752i	$0.301486\pi$
$3$	8.43597i	1.62350i	−0.584003	+	0.811752i	$0.698514\pi$
$4$	0	0
$5$	12.2748i	1.09789i	0.835858	+	0.548945i	$0.184971\pi$
$5$	12.2748i	1.09789i	−0.835858	+	0.548945i	$0.815029\pi$
$6$	0	0
$7$	1.63924	0.0885109	0.0442554	−	0.999020i	$-0.485908\pi$
$7$	1.63924	0.0885109	0.0442554	+	0.999020i	$0.485908\pi$
$8$	0	0
$9$	−44.1656	−1.63576
$10$	0	0
$11$	25.7416i	0.705580i	0.935702	+	0.352790i	$0.114767\pi$
$11$	25.7416i	0.705580i	−0.935702	+	0.352790i	$0.885233\pi$
$12$	0	0
$13$	13.2187i	0.282015i	0.990009	+	0.141008i	$0.0450342\pi$
$13$	13.2187i	0.282015i	−0.990009	+	0.141008i	$0.954966\pi$
$14$	0	0
$15$	−103.550	−1.78243
$16$	0	0
$17$	−53.6113	−0.764862	−0.382431	−	0.923984i	$-0.624913\pi$
$17$	−53.6113	−0.764862	−0.382431	+	0.923984i	$0.624913\pi$
$18$	0	0
$19$	100.391i	1.21217i	0.795400	+	0.606085i	$0.207261\pi$
$19$	100.391i	1.21217i	−0.795400	+	0.606085i	$0.792739\pi$
$20$	0	0
$21$	13.8286i	0.143698i
$22$	0	0
$23$	25.1189	0.227724	0.113862	−	0.993497i	$-0.463678\pi$
$23$	25.1189	0.227724	0.113862	+	0.993497i	$0.463678\pi$
$24$	0	0
$25$	−25.6706	−0.205365
$26$	0	0
$27$	− 144.809i	− 1.03216i
$28$	0	0
$29$	− 256.105i	− 1.63992i	−0.572424	−	0.819958i	$-0.693997\pi$
$29$	− 256.105i	− 1.63992i	0.572424	−	0.819958i	$-0.306003\pi$
$30$	0	0
$31$	−132.684	−0.768733	−0.384367	−	0.923180i	$-0.625580\pi$
$31$	−132.684	−0.768733	−0.384367	+	0.923180i	$0.625580\pi$
$32$	0	0
$33$	−217.155	−1.14551
$34$	0	0
$35$	20.1214i	0.0971753i
$36$	0	0
$37$	247.306i	1.09884i	0.835548	+	0.549418i	$0.185151\pi$
$37$	247.306i	1.09884i	−0.835548	+	0.549418i	$0.814849\pi$
$38$	0	0
$39$	−111.512	−0.457853
$40$	0	0
$41$	−198.660	−0.756720	−0.378360	−	0.925658i	$-0.623512\pi$
$41$	−198.660	−0.756720	−0.378360	+	0.925658i	$0.623512\pi$
$42$	0	0
$43$	− 404.064i	− 1.43300i	−0.697585	−	0.716502i	$-0.745742\pi$
$43$	− 404.064i	− 1.43300i	0.697585	−	0.716502i	$-0.254258\pi$
$44$	0	0
$45$	− 542.124i	− 1.79589i
$46$	0	0
$47$	78.3629	0.243200	0.121600	−	0.992579i	$-0.461197\pi$
$47$	78.3629	0.243200	0.121600	+	0.992579i	$0.461197\pi$
$48$	0	0
$49$	−340.313	−0.992166
$50$	0	0
$51$	− 452.264i	− 1.24176i
$52$	0	0
$53$	743.559i	1.92709i	0.267549	+	0.963544i	$0.413786\pi$
$53$	743.559i	1.92709i	−0.267549	+	0.963544i	$0.586214\pi$
$54$	0	0
$55$	−315.973	−0.774650
$56$	0	0
$57$	−846.894	−1.96796
$58$	0	0
$59$	− 65.8036i	− 0.145202i	−0.997361	−	0.0726008i	$-0.976870\pi$
$59$	− 65.8036i	− 0.145202i	0.997361	−	0.0726008i	$-0.0231299\pi$
$60$	0	0
$61$	273.392i	0.573841i	0.957954	+	0.286921i	$0.0926316\pi$
$61$	273.392i	0.573841i	−0.957954	+	0.286921i	$0.907368\pi$
$62$	0	0
$63$	−72.3982	−0.144783
$64$	0	0
$65$	−162.256	−0.309622
$66$	0	0
$67$	− 399.066i	− 0.727667i	−0.931464	−	0.363834i	$-0.881468\pi$
$67$	− 399.066i	− 0.727667i	0.931464	−	0.363834i	$-0.118532\pi$
$68$	0	0
$69$	211.903i	0.369711i
$70$	0	0
$71$	727.536	1.21609	0.608046	−	0.793901i	$-0.291953\pi$
$71$	727.536	1.21609	0.608046	+	0.793901i	$0.291953\pi$
$72$	0	0
$73$	−106.065	−0.170054	−0.0850270	−	0.996379i	$-0.527098\pi$
$73$	−106.065	−0.170054	−0.0850270	+	0.996379i	$0.527098\pi$
$74$	0	0
$75$	− 216.556i	− 0.333410i
$76$	0	0
$77$	42.1968i	0.0624515i
$78$	0	0
$79$	−58.9970	−0.0840213	−0.0420107	−	0.999117i	$-0.513376\pi$
$79$	−58.9970	−0.0840213	−0.0420107	+	0.999117i	$0.513376\pi$
$80$	0	0
$81$	29.1298	0.0399586
$82$	0	0
$83$	− 580.049i	− 0.767092i	−0.923522	−	0.383546i	$-0.874703\pi$
$83$	− 580.049i	− 0.767092i	0.923522	−	0.383546i	$-0.125297\pi$
$84$	0	0
$85$	− 658.068i	− 0.839735i
$86$	0	0
$87$	2160.50	2.66241
$88$	0	0
$89$	768.959	0.915837	0.457918	−	0.888994i	$-0.348595\pi$
$89$	768.959	0.915837	0.457918	+	0.888994i	$0.348595\pi$
$90$	0	0
$91$	21.6686i	0.0249614i
$92$	0	0
$93$	− 1119.32i	− 1.24804i
$94$	0	0
$95$	−1232.28	−1.33083
$96$	0	0
$97$	−809.953	−0.847817	−0.423908	−	0.905705i	$-0.639342\pi$
$97$	−809.953	−0.847817	−0.423908	+	0.905705i	$0.639342\pi$
$98$	0	0
$99$	− 1136.89i	− 1.15416i
$100$	0	0
$101$	− 428.775i	− 0.422422i	−0.977440	−	0.211211i	$-0.932259\pi$
$101$	− 428.775i	− 0.422422i	0.977440	−	0.211211i	$-0.0677408\pi$
$102$	0	0
$103$	962.201	0.920471	0.460235	−	0.887797i	$-0.347765\pi$
$103$	962.201	0.920471	0.460235	+	0.887797i	$0.347765\pi$
$104$	0	0
$105$	−169.743	−0.157764
$106$	0	0
$107$	1030.02i	0.930618i	0.885148	+	0.465309i	$0.154057\pi$
$107$	1030.02i	0.930618i	−0.885148	+	0.465309i	$0.845943\pi$
$108$	0	0
$109$	838.993i	0.737256i	0.929577	+	0.368628i	$0.120172\pi$
$109$	838.993i	0.737256i	−0.929577	+	0.368628i	$0.879828\pi$
$110$	0	0
$111$	−2086.27	−1.78396
$112$	0	0
$113$	−351.938	−0.292987	−0.146493	−	0.989212i	$-0.546799\pi$
$113$	−351.938	−0.292987	−0.146493	+	0.989212i	$0.546799\pi$
$114$	0	0
$115$	308.330i	0.250017i
$116$	0	0
$117$	− 583.810i	− 0.461310i
$118$	0	0
$119$	−87.8821	−0.0676986
$120$	0	0
$121$	668.370	0.502157
$122$	0	0
$123$	− 1675.89i	− 1.22854i
$124$	0	0
$125$	1219.25i	0.872423i
$126$	0	0
$127$	2365.81	1.65301	0.826504	−	0.562931i	$-0.190326\pi$
$127$	2365.81	1.65301	0.826504	+	0.562931i	$0.190326\pi$
$128$	0	0
$129$	3408.67	2.32649
$130$	0	0
$131$	− 570.570i	− 0.380541i	−0.981732	−	0.190271i	$-0.939063\pi$
$131$	− 570.570i	− 0.380541i	0.981732	−	0.190271i	$-0.0609365\pi$
$132$	0	0
$133$	164.565i	0.107290i
$134$	0	0
$135$	1777.50	1.13320
$136$	0	0
$137$	856.850	0.534348	0.267174	−	0.963648i	$-0.413910\pi$
$137$	856.850	0.534348	0.267174	+	0.963648i	$0.413910\pi$
$138$	0	0
$139$	− 2389.08i	− 1.45783i	−0.684603	−	0.728916i	$-0.740024\pi$
$139$	− 2389.08i	− 1.45783i	0.684603	−	0.728916i	$-0.259976\pi$
$140$	0	0
$141$	661.067i	0.394836i
$142$	0	0
$143$	−340.269	−0.198984
$144$	0	0
$145$	3143.64	1.80045
$146$	0	0
$147$	− 2870.87i	− 1.61078i
$148$	0	0
$149$	45.5671i	0.0250537i	0.999922	+	0.0125269i	$0.00398753\pi$
$149$	45.5671i	0.0250537i	−0.999922	+	0.0125269i	$0.996012\pi$
$150$	0	0
$151$	1077.06	0.580460	0.290230	−	0.956957i	$-0.406268\pi$
$151$	1077.06	0.580460	0.290230	+	0.956957i	$0.406268\pi$
$152$	0	0
$153$	2367.78	1.25113
$154$	0	0
$155$	− 1628.67i	− 0.843986i
$156$	0	0
$157$	2694.94i	1.36994i	0.728573	+	0.684968i	$0.240184\pi$
$157$	2694.94i	1.36994i	−0.728573	+	0.684968i	$0.759816\pi$
$158$	0	0
$159$	−6272.64	−3.12863
$160$	0	0
$161$	41.1761	0.0201561
$162$	0	0
$163$	− 1003.85i	− 0.482377i	−0.970478	−	0.241189i	$-0.922463\pi$
$163$	− 1003.85i	− 0.482377i	0.970478	−	0.241189i	$-0.0775372\pi$
$164$	0	0
$165$	− 2665.54i	− 1.25765i
$166$	0	0
$167$	−3460.66	−1.60356	−0.801778	−	0.597623i	$-0.796112\pi$
$167$	−3460.66	−1.60356	−0.801778	+	0.597623i	$0.796112\pi$
$168$	0	0
$169$	2022.27	0.920467
$170$	0	0
$171$	− 4433.82i	− 1.98282i
$172$	0	0
$173$	1843.49i	0.810162i	0.914281	+	0.405081i	$0.132757\pi$
$173$	1843.49i	0.810162i	−0.914281	+	0.405081i	$0.867243\pi$
$174$	0	0
$175$	−42.0803	−0.0181770
$176$	0	0
$177$	555.117	0.235735
$178$	0	0
$179$	− 1530.67i	− 0.639149i	−0.947561	−	0.319575i	$-0.896460\pi$
$179$	− 1530.67i	− 0.639149i	0.947561	−	0.319575i	$-0.103540\pi$
$180$	0	0
$181$	− 4163.35i	− 1.70972i	−0.518860	−	0.854859i	$-0.673643\pi$
$181$	− 4163.35i	− 1.70972i	0.518860	−	0.854859i	$-0.326357\pi$
$182$	0	0
$183$	−2306.33	−0.931633
$184$	0	0
$185$	−3035.64	−1.20640
$186$	0	0
$187$	− 1380.04i	− 0.539672i
$188$	0	0
$189$	− 237.377i	− 0.0913577i
$190$	0	0
$191$	430.650	0.163145	0.0815726	−	0.996667i	$-0.474006\pi$
$191$	430.650	0.163145	0.0815726	+	0.996667i	$0.474006\pi$
$192$	0	0
$193$	−2266.98	−0.845497	−0.422749	−	0.906247i	$-0.638935\pi$
$193$	−2266.98	−0.845497	−0.422749	+	0.906247i	$0.638935\pi$
$194$	0	0
$195$	− 1368.79i	− 0.502672i
$196$	0	0
$197$	− 1469.95i	− 0.531621i	−0.964025	−	0.265810i	$-0.914360\pi$
$197$	− 1469.95i	− 0.531621i	0.964025	−	0.265810i	$-0.0856395\pi$
$198$	0	0
$199$	4989.44	1.77735	0.888674	−	0.458540i	$-0.151627\pi$
$199$	4989.44	1.77735	0.888674	+	0.458540i	$0.151627\pi$
$200$	0	0
$201$	3366.51	1.18137
$202$	0	0
$203$	− 419.819i	− 0.145150i
$204$	0	0
$205$	− 2438.51i	− 0.830796i
$206$	0	0
$207$	−1109.39	−0.372503
$208$	0	0
$209$	−2584.22	−0.855283
$210$	0	0
$211$	3749.79i	1.22344i	0.791074	+	0.611720i	$0.209522\pi$
$211$	3749.79i	1.22344i	−0.791074	+	0.611720i	$0.790478\pi$
$212$	0	0
$213$	6137.47i	1.97433i
$214$	0	0
$215$	4959.80	1.57328
$216$	0	0
$217$	−217.501	−0.0680413
$218$	0	0
$219$	− 894.760i	− 0.276083i
$220$	0	0
$221$	− 708.670i	− 0.215703i
$222$	0	0
$223$	−3690.85	−1.10833	−0.554165	−	0.832407i	$-0.686962\pi$
$223$	−3690.85	−1.10833	−0.554165	+	0.832407i	$0.686962\pi$
$224$	0	0
$225$	1133.76	0.335928
$226$	0	0
$227$	2418.90i	0.707259i	0.935385	+	0.353630i	$0.115053\pi$
$227$	2418.90i	0.707259i	−0.935385	+	0.353630i	$0.884947\pi$
$228$	0	0
$229$	− 129.827i	− 0.0374637i	−0.999825	−	0.0187318i	$-0.994037\pi$
$229$	− 129.827i	− 0.0374637i	0.999825	−	0.0187318i	$-0.00596288\pi$
$230$	0	0
$231$	−355.971	−0.101390
$232$	0	0
$233$	−4259.71	−1.19769	−0.598847	−	0.800863i	$-0.704374\pi$
$233$	−4259.71	−1.19769	−0.598847	+	0.800863i	$0.704374\pi$
$234$	0	0
$235$	961.889i	0.267007i
$236$	0	0
$237$	− 497.697i	− 0.136409i
$238$	0	0
$239$	−5053.12	−1.36761	−0.683806	−	0.729664i	$-0.739676\pi$
$239$	−5053.12	−1.36761	−0.683806	+	0.729664i	$0.739676\pi$
$240$	0	0
$241$	−48.8379	−0.0130536	−0.00652681	−	0.999979i	$-0.502078\pi$
$241$	−48.8379	−0.0130536	−0.00652681	+	0.999979i	$0.502078\pi$
$242$	0	0
$243$	− 3664.09i	− 0.967291i
$244$	0	0
$245$	− 4177.27i	− 1.08929i
$246$	0	0
$247$	−1327.03	−0.341850
$248$	0	0
$249$	4893.28	1.24538
$250$	0	0
$251$	3683.69i	0.926345i	0.886268	+	0.463172i	$0.153289\pi$
$251$	3683.69i	0.926345i	−0.886268	+	0.463172i	$0.846711\pi$
$252$	0	0
$253$	646.602i	0.160678i
$254$	0	0
$255$	5551.44	1.36331
$256$	0	0
$257$	739.054	0.179381	0.0896905	−	0.995970i	$-0.471412\pi$
$257$	739.054	0.179381	0.0896905	+	0.995970i	$0.471412\pi$
$258$	0	0
$259$	405.396i	0.0972589i
$260$	0	0
$261$	11311.0i	2.68251i
$262$	0	0
$263$	2448.30	0.574025	0.287012	−	0.957927i	$-0.407338\pi$
$263$	2448.30	0.574025	0.287012	+	0.957927i	$0.407338\pi$
$264$	0	0
$265$	−9127.03	−2.11573
$266$	0	0
$267$	6486.91i	1.48686i
$268$	0	0
$269$	1173.73i	0.266035i	0.991114	+	0.133018i	$0.0424667\pi$
$269$	1173.73i	0.266035i	−0.991114	+	0.133018i	$0.957533\pi$
$270$	0	0
$271$	1404.85	0.314902	0.157451	−	0.987527i	$-0.449672\pi$
$271$	1404.85	0.314902	0.157451	+	0.987527i	$0.449672\pi$
$272$	0	0
$273$	−182.796	−0.0405249
$274$	0	0
$275$	− 660.801i	− 0.144901i
$276$	0	0
$277$	3175.88i	0.688881i	0.938808	+	0.344440i	$0.111931\pi$
$277$	3175.88i	0.688881i	−0.938808	+	0.344440i	$0.888069\pi$
$278$	0	0
$279$	5860.07	1.25747
$280$	0	0
$281$	−6045.97	−1.28353	−0.641766	−	0.766900i	$-0.721798\pi$
$281$	−6045.97	−1.28353	−0.641766	+	0.766900i	$0.721798\pi$
$282$	0	0
$283$	− 3478.45i	− 0.730644i	−0.930881	−	0.365322i	$-0.880959\pi$
$283$	− 3478.45i	− 0.730644i	0.930881	−	0.365322i	$-0.119041\pi$
$284$	0	0
$285$	− 10395.4i	− 2.16061i
$286$	0	0
$287$	−325.653	−0.0669780
$288$	0	0
$289$	−2038.82	−0.414986
$290$	0	0
$291$	− 6832.74i	− 1.37643i
$292$	0	0
$293$	− 2619.82i	− 0.522360i	−0.965290	−	0.261180i	$-0.915888\pi$
$293$	− 2619.82i	− 0.522360i	0.965290	−	0.261180i	$-0.0841116\pi$
$294$	0	0
$295$	807.725	0.159415
$296$	0	0
$297$	3727.60	0.728275
$298$	0	0
$299$	332.039i	0.0642217i
$300$	0	0
$301$	− 662.360i	− 0.126837i
$302$	0	0
$303$	3617.13	0.685804
$304$	0	0
$305$	−3355.83	−0.630015
$306$	0	0
$307$	2980.24i	0.554044i	0.960864	+	0.277022i	$0.0893475\pi$
$307$	2980.24i	0.554044i	−0.960864	+	0.277022i	$0.910653\pi$
$308$	0	0
$309$	8117.10i	1.49439i
$310$	0	0
$311$	−5294.90	−0.965422	−0.482711	−	0.875780i	$-0.660348\pi$
$311$	−5294.90	−0.965422	−0.482711	+	0.875780i	$0.660348\pi$
$312$	0	0
$313$	−4005.87	−0.723403	−0.361702	−	0.932294i	$-0.617804\pi$
$313$	−4005.87	−0.723403	−0.361702	+	0.932294i	$0.617804\pi$
$314$	0	0
$315$	− 888.673i	− 0.158956i
$316$	0	0
$317$	1144.44i	0.202770i	0.994847	+	0.101385i	$0.0323274\pi$
$317$	1144.44i	0.202770i	−0.994847	+	0.101385i	$0.967673\pi$
$318$	0	0
$319$	6592.56	1.15709
$320$	0	0
$321$	−8689.25	−1.51086
$322$	0	0
$323$	− 5382.08i	− 0.927143i
$324$	0	0
$325$	− 339.331i	− 0.0579159i
$326$	0	0
$327$	−7077.72	−1.19694
$328$	0	0
$329$	128.456	0.0215259
$330$	0	0
$331$	− 5981.64i	− 0.993295i	−0.867952	−	0.496648i	$-0.834564\pi$
$331$	− 5981.64i	− 0.993295i	0.867952	−	0.496648i	$-0.165436\pi$
$332$	0	0
$333$	− 10922.4i	− 1.79744i
$334$	0	0
$335$	4898.46	0.798899
$336$	0	0
$337$	−10002.6	−1.61684	−0.808419	−	0.588607i	$-0.799677\pi$
$337$	−10002.6	−1.61684	−0.808419	+	0.588607i	$0.799677\pi$
$338$	0	0
$339$	− 2968.94i	− 0.475665i
$340$	0	0
$341$	− 3415.50i	− 0.542403i
$342$	0	0
$343$	−1120.12	−0.176328
$344$	0	0
$345$	−2601.06	−0.405903
$346$	0	0
$347$	9064.38i	1.40231i	0.713009	+	0.701155i	$0.247332\pi$
$347$	9064.38i	1.40231i	−0.713009	+	0.701155i	$0.752668\pi$
$348$	0	0
$349$	− 7782.74i	− 1.19370i	−0.802354	−	0.596849i	$-0.796419\pi$
$349$	− 7782.74i	− 1.19370i	0.802354	−	0.596849i	$-0.203581\pi$
$350$	0	0
$351$	1914.18	0.291086
$352$	0	0
$353$	−1411.35	−0.212800	−0.106400	−	0.994323i	$-0.533932\pi$
$353$	−1411.35	−0.212800	−0.106400	+	0.994323i	$0.533932\pi$
$354$	0	0
$355$	8930.35i	1.33514i
$356$	0	0
$357$	− 741.371i	− 0.109909i
$358$	0	0
$359$	−2160.73	−0.317658	−0.158829	−	0.987306i	$-0.550772\pi$
$359$	−2160.73	−0.317658	−0.158829	+	0.987306i	$0.550772\pi$
$360$	0	0
$361$	−3219.31	−0.469355
$362$	0	0
$363$	5638.35i	0.815253i
$364$	0	0
$365$	− 1301.92i	− 0.186701i
$366$	0	0
$367$	10757.7	1.53010	0.765052	−	0.643969i	$-0.222713\pi$
$367$	10757.7	1.53010	0.765052	+	0.643969i	$0.222713\pi$
$368$	0	0
$369$	8773.95	1.23782
$370$	0	0
$371$	1218.87i	0.170568i
$372$	0	0
$373$	− 1989.79i	− 0.276213i	−0.990417	−	0.138107i	$-0.955898\pi$
$373$	− 1989.79i	− 0.276213i	0.990417	−	0.138107i	$-0.0441016\pi$
$374$	0	0
$375$	−10285.5	−1.41638
$376$	0	0
$377$	3385.37	0.462481
$378$	0	0
$379$	1622.04i	0.219838i	0.993941	+	0.109919i	$0.0350591\pi$
$379$	1622.04i	0.219838i	−0.993941	+	0.109919i	$0.964941\pi$
$380$	0	0
$381$	19957.9i	2.68366i
$382$	0	0
$383$	−9042.17	−1.20635	−0.603176	−	0.797608i	$-0.706098\pi$
$383$	−9042.17	−1.20635	−0.603176	+	0.797608i	$0.706098\pi$
$384$	0	0
$385$	−517.957	−0.0685650
$386$	0	0
$387$	17845.7i	2.34406i
$388$	0	0
$389$	3642.08i	0.474706i	0.971423	+	0.237353i	$0.0762799\pi$
$389$	3642.08i	0.474706i	−0.971423	+	0.237353i	$0.923720\pi$
$390$	0	0
$391$	−1346.66	−0.174178
$392$	0	0
$393$	4813.31	0.617810
$394$	0	0
$395$	− 724.176i	− 0.0922462i
$396$	0	0
$397$	10071.3i	1.27320i	0.771192	+	0.636602i	$0.219661\pi$
$397$	10071.3i	1.27320i	−0.771192	+	0.636602i	$0.780339\pi$
$398$	0	0
$399$	−1388.27	−0.174186
$400$	0	0
$401$	3025.14	0.376729	0.188365	−	0.982099i	$-0.439681\pi$
$401$	3025.14	0.376729	0.188365	+	0.982099i	$0.439681\pi$
$402$	0	0
$403$	− 1753.90i	− 0.216794i
$404$	0	0
$405$	357.562i	0.0438702i
$406$	0	0
$407$	−6366.06	−0.775317
$408$	0	0
$409$	−9440.21	−1.14129	−0.570646	−	0.821196i	$-0.693307\pi$
$409$	−9440.21	−1.14129	−0.570646	+	0.821196i	$0.693307\pi$
$410$	0	0
$411$	7228.36i	0.867515i
$412$	0	0
$413$	− 107.868i	− 0.0128519i
$414$	0	0
$415$	7119.98	0.842183
$416$	0	0
$417$	20154.2	2.36680
$418$	0	0
$419$	4604.25i	0.536831i	0.963303	+	0.268416i	$0.0865000\pi$
$419$	4604.25i	0.536831i	−0.963303	+	0.268416i	$0.913500\pi$
$420$	0	0
$421$	− 13347.4i	− 1.54516i	−0.634917	−	0.772580i	$-0.718966\pi$
$421$	− 13347.4i	− 1.54516i	0.634917	−	0.772580i	$-0.281034\pi$
$422$	0	0
$423$	−3460.95	−0.397818
$424$	0	0
$425$	1376.23	0.157076
$426$	0	0
$427$	448.157i	0.0507912i
$428$	0	0
$429$	− 2870.50i	− 0.323052i
$430$	0	0
$431$	−10617.7	−1.18663	−0.593314	−	0.804971i	$-0.702181\pi$
$431$	−10617.7	−1.18663	−0.593314	+	0.804971i	$0.702181\pi$
$432$	0	0
$433$	−706.479	−0.0784093	−0.0392046	−	0.999231i	$-0.512482\pi$
$433$	−706.479	−0.0784093	−0.0392046	+	0.999231i	$0.512482\pi$
$434$	0	0
$435$	26519.7i	2.92303i
$436$	0	0
$437$	2521.71i	0.276041i
$438$	0	0
$439$	13611.8	1.47985	0.739926	−	0.672688i	$-0.234860\pi$
$439$	13611.8	1.47985	0.739926	+	0.672688i	$0.234860\pi$
$440$	0	0
$441$	15030.1	1.62295
$442$	0	0
$443$	− 4422.21i	− 0.474279i	−0.971476	−	0.237139i	$-0.923790\pi$
$443$	− 4422.21i	− 0.474279i	0.971476	−	0.237139i	$-0.0762098\pi$
$444$	0	0
$445$	9438.81i	1.00549i
$446$	0	0
$447$	−384.403	−0.0406748
$448$	0	0
$449$	−5231.76	−0.549893	−0.274947	−	0.961460i	$-0.588660\pi$
$449$	−5231.76	−0.549893	−0.274947	+	0.961460i	$0.588660\pi$
$450$	0	0
$451$	− 5113.83i	− 0.533927i
$452$	0	0
$453$	9086.01i	0.942379i
$454$	0	0
$455$	−265.978	−0.0274049
$456$	0	0
$457$	6833.10	0.699429	0.349715	−	0.936856i	$-0.386279\pi$
$457$	6833.10	0.699429	0.349715	+	0.936856i	$0.386279\pi$
$458$	0	0
$459$	7763.38i	0.789463i
$460$	0	0
$461$	8451.19i	0.853821i	0.904294	+	0.426910i	$0.140398\pi$
$461$	8451.19i	0.853821i	−0.904294	+	0.426910i	$0.859602\pi$
$462$	0	0
$463$	−4273.38	−0.428943	−0.214472	−	0.976730i	$-0.568803\pi$
$463$	−4273.38	−0.428943	−0.214472	+	0.976730i	$0.568803\pi$
$464$	0	0
$465$	13739.4	1.37021
$466$	0	0
$467$	17317.9i	1.71601i	0.513643	+	0.858004i	$0.328296\pi$
$467$	17317.9i	1.71601i	−0.513643	+	0.858004i	$0.671704\pi$
$468$	0	0
$469$	− 654.167i	− 0.0644065i
$470$	0	0
$471$	−22734.5	−2.22410
$472$	0	0
$473$	10401.3	1.01110
$474$	0	0
$475$	− 2577.09i	− 0.248937i
$476$	0	0
$477$	− 32839.7i	− 3.15226i
$478$	0	0
$479$	4067.97	0.388038	0.194019	−	0.980998i	$-0.437848\pi$
$479$	4067.97	0.388038	0.194019	+	0.980998i	$0.437848\pi$
$480$	0	0
$481$	−3269.06	−0.309888
$482$	0	0
$483$	347.360i	0.0327235i
$484$	0	0
$485$	− 9942.00i	− 0.930811i
$486$	0	0
$487$	−16174.3	−1.50499	−0.752493	−	0.658600i	$-0.771149\pi$
$487$	−16174.3	−1.50499	−0.752493	+	0.658600i	$0.771149\pi$
$488$	0	0
$489$	8468.44	0.783141
$490$	0	0
$491$	19228.6i	1.76736i	0.468091	+	0.883680i	$0.344942\pi$
$491$	19228.6i	1.76736i	−0.468091	+	0.883680i	$0.655058\pi$
$492$	0	0
$493$	13730.1i	1.25431i
$494$	0	0
$495$	13955.1	1.26714
$496$	0	0
$497$	1192.61	0.107637
$498$	0	0
$499$	− 20713.6i	− 1.85825i	−0.369762	−	0.929127i	$-0.620561\pi$
$499$	− 20713.6i	− 1.85825i	0.369762	−	0.929127i	$-0.379439\pi$
$500$	0	0
$501$	− 29194.0i	− 2.60338i
$502$	0	0
$503$	9828.84	0.871265	0.435632	−	0.900125i	$-0.356525\pi$
$503$	9828.84	0.871265	0.435632	+	0.900125i	$0.356525\pi$
$504$	0	0
$505$	5263.12	0.463774
$506$	0	0
$507$	17059.8i	1.49438i
$508$	0	0
$509$	19029.8i	1.65714i	0.559889	+	0.828568i	$0.310844\pi$
$509$	19029.8i	1.65714i	−0.559889	+	0.828568i	$0.689156\pi$
$510$	0	0
$511$	−173.866	−0.0150516
$512$	0	0
$513$	14537.5	1.25116
$514$	0	0
$515$	11810.8i	1.01058i
$516$	0	0
$517$	2017.19i	0.171597i
$518$	0	0
$519$	−15551.6	−1.31530
$520$	0	0
$521$	−10607.1	−0.891950	−0.445975	−	0.895045i	$-0.647143\pi$
$521$	−10607.1	−0.891950	−0.445975	+	0.895045i	$0.647143\pi$
$522$	0	0
$523$	5519.88i	0.461506i	0.973012	+	0.230753i	$0.0741190\pi$
$523$	5519.88i	0.461506i	−0.973012	+	0.230753i	$0.925881\pi$
$524$	0	0
$525$	− 354.989i	− 0.0295104i
$526$	0	0
$527$	7113.36	0.587975
$528$	0	0
$529$	−11536.0	−0.948142
$530$	0	0
$531$	2906.25i	0.237515i
$532$	0	0
$533$	− 2626.02i	− 0.213407i
$534$	0	0
$535$	−12643.3	−1.02172
$536$	0	0
$537$	12912.7	1.03766
$538$	0	0
$539$	− 8760.20i	− 0.700053i
$540$	0	0
$541$	13481.4i	1.07137i	0.844419	+	0.535683i	$0.179946\pi$
$541$	13481.4i	1.07137i	−0.844419	+	0.535683i	$0.820054\pi$
$542$	0	0
$543$	35121.9	2.77573
$544$	0	0
$545$	−10298.5	−0.809427
$546$	0	0
$547$	1743.55i	0.136287i	0.997676	+	0.0681434i	$0.0217075\pi$
$547$	1743.55i	0.136287i	−0.997676	+	0.0681434i	$0.978292\pi$
$548$	0	0
$549$	− 12074.5i	− 0.938668i
$550$	0	0
$551$	25710.6	1.98786
$552$	0	0
$553$	−96.7105	−0.00743680
$554$	0	0
$555$	− 25608.5i	− 1.95860i
$556$	0	0
$557$	4086.47i	0.310861i	0.987847	+	0.155430i	$0.0496764\pi$
$557$	4086.47i	0.310861i	−0.987847	+	0.155430i	$0.950324\pi$
$558$	0	0
$559$	5341.19	0.404129
$560$	0	0
$561$	11642.0	0.876159
$562$	0	0
$563$	99.1014i	0.00741852i	0.999993	+	0.00370926i	$0.00118070\pi$
$563$	99.1014i	0.00741852i	−0.999993	+	0.00370926i	$0.998819\pi$
$564$	0	0
$565$	− 4319.96i	− 0.321668i
$566$	0	0
$567$	47.7509	0.00353677
$568$	0	0
$569$	−8915.23	−0.656847	−0.328423	−	0.944531i	$-0.606517\pi$
$569$	−8915.23	−0.656847	−0.328423	+	0.944531i	$0.606517\pi$
$570$	0	0
$571$	− 6995.13i	− 0.512674i	−0.966587	−	0.256337i	$-0.917484\pi$
$571$	− 6995.13i	− 0.512674i	0.966587	−	0.256337i	$-0.0825157\pi$
$572$	0	0
$573$	3632.95i	0.264867i
$574$	0	0
$575$	−644.818	−0.0467665
$576$	0	0
$577$	17911.5	1.29232	0.646159	−	0.763203i	$-0.276374\pi$
$577$	17911.5	1.29232	0.646159	+	0.763203i	$0.276374\pi$
$578$	0	0
$579$	− 19124.2i	− 1.37267i
$580$	0	0
$581$	− 950.842i	− 0.0678960i
$582$	0	0
$583$	−19140.4	−1.35972
$584$	0	0
$585$	7166.15	0.506468
$586$	0	0
$587$	11229.2i	0.789573i	0.918773	+	0.394787i	$0.129181\pi$
$587$	11229.2i	0.789573i	−0.918773	+	0.394787i	$0.870819\pi$
$588$	0	0
$589$	− 13320.2i	− 0.931835i
$590$	0	0
$591$	12400.4	0.863088
$592$	0	0
$593$	7006.26	0.485181	0.242591	−	0.970129i	$-0.422003\pi$
$593$	7006.26	0.485181	0.242591	+	0.970129i	$0.422003\pi$
$594$	0	0
$595$	− 1078.73i	− 0.0743257i
$596$	0	0
$597$	42090.8i	2.88553i
$598$	0	0
$599$	−8502.74	−0.579987	−0.289994	−	0.957029i	$-0.593653\pi$
$599$	−8502.74	−0.579987	−0.289994	+	0.957029i	$0.593653\pi$
$600$	0	0
$601$	−11936.2	−0.810127	−0.405063	−	0.914289i	$-0.632751\pi$
$601$	−11936.2	−0.810127	−0.405063	+	0.914289i	$0.632751\pi$
$602$	0	0
$603$	17625.0i	1.19029i
$604$	0	0
$605$	8204.11i	0.551313i
$606$	0	0
$607$	3850.00	0.257441	0.128721	−	0.991681i	$-0.458913\pi$
$607$	3850.00	0.257441	0.128721	+	0.991681i	$0.458913\pi$
$608$	0	0
$609$	3541.58	0.235652
$610$	0	0
$611$	1035.85i	0.0685861i
$612$	0	0
$613$	− 8938.34i	− 0.588933i	−0.955662	−	0.294467i	$-0.904858\pi$
$613$	− 8938.34i	− 0.588933i	0.955662	−	0.294467i	$-0.0951420\pi$
$614$	0	0
$615$	20571.2	1.34880
$616$	0	0
$617$	2585.09	0.168674	0.0843370	−	0.996437i	$-0.473123\pi$
$617$	2585.09	0.168674	0.0843370	+	0.996437i	$0.473123\pi$
$618$	0	0
$619$	10359.1i	0.672648i	0.941746	+	0.336324i	$0.109184\pi$
$619$	10359.1i	0.672648i	−0.941746	+	0.336324i	$0.890816\pi$
$620$	0	0
$621$	− 3637.44i	− 0.235049i
$622$	0	0
$623$	1260.51	0.0810615
$624$	0	0
$625$	−18174.8	−1.16319
$626$	0	0
$627$	− 21800.4i	− 1.38855i
$628$	0	0
$629$	− 13258.4i	− 0.840458i
$630$	0	0
$631$	14411.5	0.909210	0.454605	−	0.890693i	$-0.349780\pi$
$631$	14411.5	0.909210	0.454605	+	0.890693i	$0.349780\pi$
$632$	0	0
$633$	−31633.1	−1.98626
$634$	0	0
$635$	29039.9i	1.81482i
$636$	0	0
$637$	− 4498.48i	− 0.279806i
$638$	0	0
$639$	−32132.1	−1.98924
$640$	0	0
$641$	−25724.0	−1.58508	−0.792542	−	0.609818i	$-0.791243\pi$
$641$	−25724.0	−1.58508	−0.792542	+	0.609818i	$0.791243\pi$
$642$	0	0
$643$	11081.4i	0.679640i	0.940491	+	0.339820i	$0.110366\pi$
$643$	11081.4i	0.679640i	−0.940491	+	0.339820i	$0.889634\pi$
$644$	0	0
$645$	41840.8i	2.55423i
$646$	0	0
$647$	1247.43	0.0757981	0.0378991	−	0.999282i	$-0.487933\pi$
$647$	1247.43	0.0757981	0.0378991	+	0.999282i	$0.487933\pi$
$648$	0	0
$649$	1693.89	0.102451
$650$	0	0
$651$	− 1834.84i	− 0.110465i
$652$	0	0
$653$	− 11741.1i	− 0.703621i	−0.936071	−	0.351811i	$-0.885566\pi$
$653$	− 11741.1i	− 0.703621i	0.936071	−	0.351811i	$-0.114434\pi$
$654$	0	0
$655$	7003.63	0.417793
$656$	0	0
$657$	4684.42	0.278168
$658$	0	0
$659$	2398.73i	0.141792i	0.997484	+	0.0708961i	$0.0225859\pi$
$659$	2398.73i	0.141792i	−0.997484	+	0.0708961i	$0.977414\pi$
$660$	0	0
$661$	12428.5i	0.731337i	0.930745	+	0.365669i	$0.119160\pi$
$661$	12428.5i	0.731337i	−0.930745	+	0.365669i	$0.880840\pi$
$662$	0	0
$663$	5978.32	0.350194
$664$	0	0
$665$	−2020.00	−0.117793
$666$	0	0
$667$	− 6433.09i	− 0.373449i
$668$	0	0
$669$	− 31135.9i	− 1.79938i
$670$	0	0
$671$	−7037.56	−0.404891
$672$	0	0
$673$	−23869.3	−1.36716	−0.683578	−	0.729878i	$-0.739577\pi$
$673$	−23869.3	−1.36716	−0.683578	+	0.729878i	$0.739577\pi$
$674$	0	0
$675$	3717.32i	0.211970i
$676$	0	0
$677$	− 8009.12i	− 0.454676i	−0.973816	−	0.227338i	$-0.926998\pi$
$677$	− 8009.12i	− 0.454676i	0.973816	−	0.227338i	$-0.0730022\pi$
$678$	0	0
$679$	−1327.71	−0.0750410
$680$	0	0
$681$	−20405.8	−1.14824
$682$	0	0
$683$	− 12944.0i	− 0.725167i	−0.931951	−	0.362583i	$-0.881895\pi$
$683$	− 12944.0i	− 0.725167i	0.931951	−	0.362583i	$-0.118105\pi$
$684$	0	0
$685$	10517.7i	0.586655i
$686$	0	0
$687$	1095.21	0.0608224
$688$	0	0
$689$	−9828.85	−0.543468
$690$	0	0
$691$	24123.5i	1.32807i	0.747699	+	0.664037i	$0.231158\pi$
$691$	24123.5i	1.32807i	−0.747699	+	0.664037i	$0.768842\pi$
$692$	0	0
$693$	− 1863.65i	− 0.102156i
$694$	0	0
$695$	29325.4	1.60054
$696$	0	0
$697$	10650.4	0.578787
$698$	0	0
$699$	− 35934.8i	− 1.94446i
$700$	0	0
$701$	− 10918.3i	− 0.588274i	−0.955763	−	0.294137i	$-0.904968\pi$
$701$	− 10918.3i	− 0.588274i	0.955763	−	0.294137i	$-0.0950322\pi$
$702$	0	0
$703$	−24827.3	−1.33198
$704$	0	0
$705$	−8114.47	−0.433487
$706$	0	0
$707$	− 702.866i	− 0.0373890i
$708$	0	0
$709$	6473.79i	0.342917i	0.985191	+	0.171459i	$0.0548480\pi$
$709$	6473.79i	0.342917i	−0.985191	+	0.171459i	$0.945152\pi$
$710$	0	0
$711$	2605.64	0.137439
$712$	0	0
$713$	−3332.88	−0.175059
$714$	0	0
$715$	− 4176.74i	− 0.218463i
$716$	0	0
$717$	− 42628.0i	− 2.22032i
$718$	0	0
$719$	−30210.0	−1.56696	−0.783479	−	0.621418i	$-0.786557\pi$
$719$	−30210.0	−1.56696	−0.783479	+	0.621418i	$0.786557\pi$
$720$	0	0
$721$	1577.28	0.0814717
$722$	0	0
$723$	− 411.995i	− 0.0211926i
$724$	0	0
$725$	6574.37i	0.336781i
$726$	0	0
$727$	−20721.3	−1.05710	−0.528549	−	0.848903i	$-0.677264\pi$
$727$	−20721.3	−1.05710	−0.528549	+	0.848903i	$0.677264\pi$
$728$	0	0
$729$	31696.7	1.61036
$730$	0	0
$731$	21662.4i	1.09605i
$732$	0	0
$733$	− 19629.0i	− 0.989107i	−0.869147	−	0.494553i	$-0.835332\pi$
$733$	− 19629.0i	− 0.989107i	0.869147	−	0.494553i	$-0.164668\pi$
$734$	0	0
$735$	35239.3	1.76847
$736$	0	0
$737$	10272.6	0.513428
$738$	0	0
$739$	− 12436.5i	− 0.619058i	−0.950890	−	0.309529i	$-0.899829\pi$
$739$	− 12436.5i	− 0.619058i	0.950890	−	0.309529i	$-0.100171\pi$
$740$	0	0
$741$	− 11194.8i	− 0.554995i
$742$	0	0
$743$	7669.27	0.378678	0.189339	−	0.981912i	$-0.439365\pi$
$743$	7669.27	0.378678	0.189339	+	0.981912i	$0.439365\pi$
$744$	0	0
$745$	−559.327	−0.0275062
$746$	0	0
$747$	25618.2i	1.25478i
$748$	0	0
$749$	1688.46i	0.0823698i
$750$	0	0
$751$	−26531.8	−1.28916	−0.644580	−	0.764537i	$-0.722968\pi$
$751$	−26531.8	−1.28916	−0.644580	+	0.764537i	$0.722968\pi$
$752$	0	0
$753$	−31075.5	−1.50392
$754$	0	0
$755$	13220.6i	0.637282i
$756$	0	0
$757$	112.316i	0.00539258i	0.999996	+	0.00269629i	$0.000858257\pi$
$757$	112.316i	0.00539258i	−0.999996	+	0.00269629i	$0.999142\pi$
$758$	0	0
$759$	−5454.71	−0.260861
$760$	0	0
$761$	−36991.3	−1.76207	−0.881033	−	0.473055i	$-0.843151\pi$
$761$	−36991.3	−1.76207	−0.881033	+	0.473055i	$0.843151\pi$
$762$	0	0
$763$	1375.31i	0.0652552i
$764$	0	0
$765$	29064.0i	1.37361i
$766$	0	0
$767$	869.835	0.0409490
$768$	0	0
$769$	26637.0	1.24910	0.624548	−	0.780987i	$-0.285283\pi$
$769$	26637.0	1.24910	0.624548	+	0.780987i	$0.285283\pi$
$770$	0	0
$771$	6234.64i	0.291226i
$772$	0	0
$773$	27921.7i	1.29919i	0.760280	+	0.649596i	$0.225062\pi$
$773$	27921.7i	1.29919i	−0.760280	+	0.649596i	$0.774938\pi$
$774$	0	0
$775$	3406.07	0.157871
$776$	0	0
$777$	−3419.91	−0.157900
$778$	0	0
$779$	− 19943.7i	− 0.917273i
$780$	0	0
$781$	18727.9i	0.858051i
$782$	0	0
$783$	−37086.2	−1.69266
$784$	0	0
$785$	−33079.9	−1.50404
$786$	0	0
$787$	− 40839.8i	− 1.84979i	−0.380229	−	0.924893i	$-0.624155\pi$
$787$	− 40839.8i	− 1.84979i	0.380229	−	0.924893i	$-0.375845\pi$
$788$	0	0
$789$	20653.8i	0.931931i
$790$	0	0
$791$	−576.912	−0.0259325
$792$	0	0
$793$	−3613.88	−0.161832
$794$	0	0
$795$	− 76995.4i	− 3.43490i
$796$	0	0
$797$	− 23555.1i	− 1.04688i	−0.852062	−	0.523440i	$-0.824648\pi$
$797$	− 23555.1i	− 1.04688i	0.852062	−	0.523440i	$-0.175352\pi$
$798$	0	0
$799$	−4201.14	−0.186015
$800$	0	0
$801$	−33961.5	−1.49809
$802$	0	0
$803$	− 2730.28i	− 0.119987i
$804$	0	0
$805$	505.428i	0.0221292i
$806$	0	0
$807$	−9901.55	−0.431910
$808$	0	0
$809$	−34940.4	−1.51847	−0.759233	−	0.650819i	$-0.774426\pi$
$809$	−34940.4	−1.51847	−0.759233	+	0.650819i	$0.774426\pi$
$810$	0	0
$811$	− 21451.0i	− 0.928789i	−0.885628	−	0.464395i	$-0.846272\pi$
$811$	− 21451.0i	− 0.928789i	0.885628	−	0.464395i	$-0.153728\pi$
$812$	0	0
$813$	11851.3i	0.511244i
$814$	0	0
$815$	12322.0	0.529597
$816$	0	0
$817$	40564.3	1.73705
$818$	0	0
$819$	− 957.008i	− 0.0408310i
$820$	0	0
$821$	− 12318.6i	− 0.523656i	−0.965115	−	0.261828i	$-0.915675\pi$
$821$	− 12318.6i	− 0.523656i	0.965115	−	0.261828i	$-0.0843252\pi$
$822$	0	0
$823$	−24493.5	−1.03741	−0.518707	−	0.854952i	$-0.673586\pi$
$823$	−24493.5	−1.03741	−0.518707	+	0.854952i	$0.673586\pi$
$824$	0	0
$825$	5574.50	0.235248
$826$	0	0
$827$	− 37233.4i	− 1.56558i	−0.622287	−	0.782789i	$-0.713796\pi$
$827$	− 37233.4i	− 1.56558i	0.622287	−	0.782789i	$-0.286204\pi$
$828$	0	0
$829$	12881.0i	0.539658i	0.962908	+	0.269829i	$0.0869671\pi$
$829$	12881.0i	0.539658i	−0.962908	+	0.269829i	$0.913033\pi$
$830$	0	0
$831$	−26791.6	−1.11840
$832$	0	0
$833$	18244.6	0.758870
$834$	0	0
$835$	− 42478.9i	− 1.76053i
$836$	0	0
$837$	19213.8i	0.793459i
$838$	0	0
$839$	1394.89	0.0573982	0.0286991	−	0.999588i	$-0.490864\pi$
$839$	1394.89	0.0573982	0.0286991	+	0.999588i	$0.490864\pi$
$840$	0	0
$841$	−41200.9	−1.68932
$842$	0	0
$843$	− 51003.7i	− 2.08382i
$844$	0	0
$845$	24822.9i	1.01057i
$846$	0	0
$847$	1095.62	0.0444463
$848$	0	0
$849$	29344.1	1.18620
$850$	0	0
$851$	6212.08i	0.250232i
$852$	0	0
$853$	− 21615.8i	− 0.867658i	−0.900995	−	0.433829i	$-0.857162\pi$
$853$	− 21615.8i	− 0.867658i	0.900995	−	0.433829i	$-0.142838\pi$
$854$	0	0
$855$	54424.2	2.17692
$856$	0	0
$857$	2273.70	0.0906277	0.0453139	−	0.998973i	$-0.485571\pi$
$857$	2273.70	0.0906277	0.0453139	+	0.998973i	$0.485571\pi$
$858$	0	0
$859$	30652.1i	1.21750i	0.793361	+	0.608752i	$0.208329\pi$
$859$	30652.1i	1.21750i	−0.793361	+	0.608752i	$0.791671\pi$
$860$	0	0
$861$	− 2747.20i	− 0.108739i
$862$	0	0
$863$	−23721.7	−0.935686	−0.467843	−	0.883812i	$-0.654969\pi$
$863$	−23721.7	−0.935686	−0.467843	+	0.883812i	$0.654969\pi$
$864$	0	0
$865$	−22628.5	−0.889469
$866$	0	0
$867$	− 17199.5i	− 0.673731i
$868$	0	0
$869$	− 1518.68i	− 0.0592838i
$870$	0	0
$871$	5275.12	0.205213
$872$	0	0
$873$	35772.1	1.38683
$874$	0	0
$875$	1998.65i	0.0772189i
$876$	0	0
$877$	− 31720.2i	− 1.22134i	−0.791885	−	0.610670i	$-0.790900\pi$
$877$	− 31720.2i	− 1.22134i	0.791885	−	0.610670i	$-0.209100\pi$
$878$	0	0
$879$	22100.7	0.848054
$880$	0	0
$881$	−24603.0	−0.940859	−0.470429	−	0.882438i	$-0.655901\pi$
$881$	−24603.0	−0.940859	−0.470429	+	0.882438i	$0.655901\pi$
$882$	0	0
$883$	− 33215.2i	− 1.26589i	−0.774197	−	0.632945i	$-0.781846\pi$
$883$	− 33215.2i	− 1.26589i	0.774197	−	0.632945i	$-0.218154\pi$
$884$	0	0
$885$	6813.95i	0.258812i
$886$	0	0
$887$	39722.9	1.50368	0.751841	−	0.659345i	$-0.229166\pi$
$887$	39722.9	1.50368	0.751841	+	0.659345i	$0.229166\pi$
$888$	0	0
$889$	3878.15	0.146309
$890$	0	0
$891$	749.848i	0.0281940i
$892$	0	0
$893$	7866.92i	0.294800i
$894$	0	0
$895$	18788.7	0.701716
$896$	0	0
$897$	−2801.07	−0.104264
$898$	0	0
$899$	33981.1i	1.26066i
$900$	0	0
$901$	− 39863.2i	− 1.47396i
$902$	0	0
$903$	5587.65	0.205920
$904$	0	0
$905$	51104.2	1.87708
$906$	0	0
$907$	6456.50i	0.236367i	0.992992	+	0.118183i	$0.0377071\pi$
$907$	6456.50i	0.236367i	−0.992992	+	0.118183i	$0.962293\pi$
$908$	0	0
$909$	18937.1i	0.690983i
$910$	0	0
$911$	−2013.95	−0.0732438	−0.0366219	−	0.999329i	$-0.511660\pi$
$911$	−2013.95	−0.0732438	−0.0366219	+	0.999329i	$0.511660\pi$
$912$	0	0
$913$	14931.4	0.541245
$914$	0	0
$915$	− 28309.7i	− 1.02283i
$916$	0	0
$917$	− 935.303i	− 0.0336820i
$918$	0	0
$919$	−37746.5	−1.35489	−0.677443	−	0.735575i	$-0.736912\pi$
$919$	−37746.5	−1.35489	−0.677443	+	0.735575i	$0.736912\pi$
$920$	0	0
$921$	−25141.2	−0.899492
$922$	0	0
$923$	9617.05i	0.342957i
$924$	0	0
$925$	− 6348.50i	− 0.225662i
$926$	0	0
$927$	−42496.2	−1.50567
$928$	0	0
$929$	45643.5	1.61197	0.805983	−	0.591939i	$-0.201637\pi$
$929$	45643.5	1.61197	0.805983	+	0.591939i	$0.201637\pi$
$930$	0	0
$931$	− 34164.3i	− 1.20267i
$932$	0	0
$933$	− 44667.6i	− 1.56737i
$934$	0	0
$935$	16939.7	0.592501
$936$	0	0
$937$	47317.5	1.64973	0.824864	−	0.565331i	$-0.191252\pi$
$937$	47317.5	1.64973	0.824864	+	0.565331i	$0.191252\pi$
$938$	0	0
$939$	− 33793.4i	− 1.17445i
$940$	0	0
$941$	− 21318.9i	− 0.738550i	−0.929320	−	0.369275i	$-0.879606\pi$
$941$	− 21318.9i	− 0.738550i	0.929320	−	0.369275i	$-0.120394\pi$
$942$	0	0
$943$	−4990.14	−0.172324
$944$	0	0
$945$	2913.75	0.100301
$946$	0	0
$947$	20601.8i	0.706937i	0.935446	+	0.353468i	$0.114998\pi$
$947$	20601.8i	0.706937i	−0.935446	+	0.353468i	$0.885002\pi$
$948$	0	0
$949$	− 1402.03i	− 0.0479578i
$950$	0	0
$951$	−9654.45	−0.329198
$952$	0	0
$953$	42987.2	1.46117	0.730583	−	0.682824i	$-0.239248\pi$
$953$	42987.2	1.46117	0.730583	+	0.682824i	$0.239248\pi$
$954$	0	0
$955$	5286.14i	0.179116i
$956$	0	0
$957$	55614.6i	1.87854i
$958$	0	0
$959$	1404.59	0.0472956
$960$	0	0
$961$	−12186.0	−0.409049
$962$	0	0
$963$	− 45491.6i	− 1.52227i
$964$	0	0
$965$	− 27826.7i	− 0.928264i
$966$	0	0
$967$	44030.7	1.46425	0.732126	−	0.681170i	$-0.238528\pi$
$967$	44030.7	1.46425	0.732126	+	0.681170i	$0.238528\pi$
$968$	0	0
$969$	45403.1	1.50522
$970$	0	0
$971$	50487.1i	1.66860i	0.551313	+	0.834298i	$0.314127\pi$
$971$	50487.1i	1.66860i	−0.551313	+	0.834298i	$0.685873\pi$
$972$	0	0
$973$	− 3916.28i	− 0.129034i
$974$	0	0
$975$	2862.58	0.0940267
$976$	0	0
$977$	49515.3	1.62143	0.810714	−	0.585442i	$-0.199079\pi$
$977$	49515.3	1.62143	0.810714	+	0.585442i	$0.199079\pi$
$978$	0	0
$979$	19794.2i	0.646196i
$980$	0	0
$981$	− 37054.6i	− 1.20598i
$982$	0	0
$983$	40046.2	1.29936	0.649682	−	0.760206i	$-0.274902\pi$
$983$	40046.2	1.29936	0.649682	+	0.760206i	$0.274902\pi$
$984$	0	0
$985$	18043.3	0.583662
$986$	0	0
$987$	1083.65i	0.0349473i
$988$	0	0
$989$	− 10149.7i	− 0.326330i
$990$	0	0
$991$	−18673.2	−0.598560	−0.299280	−	0.954165i	$-0.596746\pi$
$991$	−18673.2	−0.598560	−0.299280	+	0.954165i	$0.596746\pi$
$992$	0	0
$993$	50460.9	1.61262
$994$	0	0
$995$	61244.4i	1.95133i
$996$	0	0
$997$	31087.3i	0.987508i	0.869602	+	0.493754i	$0.164376\pi$
$997$	31087.3i	0.987508i	−0.869602	+	0.493754i	$0.835624\pi$
$998$	0	0
$999$	35812.1	1.13418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.b.k.513.10		10
4.3	odd	2		1024.4.b.j.513.1		10
8.3	odd	2		1024.4.b.j.513.10		10
8.5	even	2	inner	1024.4.b.k.513.1		10
16.3	odd	4		1024.4.a.n.1.10		10
16.5	even	4		1024.4.a.m.1.10		10
16.11	odd	4		1024.4.a.n.1.1		10
16.13	even	4		1024.4.a.m.1.1		10
32.3	odd	8		128.4.e.b.97.5		10
32.5	even	8		128.4.e.a.33.1		10
32.11	odd	8		16.4.e.a.13.2	yes	10
32.13	even	8		64.4.e.a.49.5		10
32.19	odd	8		16.4.e.a.5.2	✓	10
32.21	even	8		64.4.e.a.17.5		10
32.27	odd	8		128.4.e.b.33.5		10
32.29	even	8		128.4.e.a.97.1		10
96.11	even	8		144.4.k.a.109.4		10
96.53	odd	8		576.4.k.a.145.1		10
96.77	odd	8		576.4.k.a.433.1		10
96.83	even	8		144.4.k.a.37.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.2	✓	10	32.19	odd	8
16.4.e.a.13.2	yes	10	32.11	odd	8
64.4.e.a.17.5		10	32.21	even	8
64.4.e.a.49.5		10	32.13	even	8
128.4.e.a.33.1		10	32.5	even	8
128.4.e.a.97.1		10	32.29	even	8
128.4.e.b.33.5		10	32.27	odd	8
128.4.e.b.97.5		10	32.3	odd	8
144.4.k.a.37.4		10	96.83	even	8
144.4.k.a.109.4		10	96.11	even	8
576.4.k.a.145.1		10	96.53	odd	8
576.4.k.a.433.1		10	96.77	odd	8
1024.4.a.m.1.1		10	16.13	even	4
1024.4.a.m.1.10		10	16.5	even	4
1024.4.a.n.1.1		10	16.11	odd	4
1024.4.a.n.1.10		10	16.3	odd	4
1024.4.b.j.513.1		10	4.3	odd	2
1024.4.b.j.513.10		10	8.3	odd	2
1024.4.b.k.513.1		10	8.5	even	2	inner
1024.4.b.k.513.10		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.43597i q^{3} +12.2748i q^{5} +1.63924 q^{7} -44.1656 q^{9} +O(q^{10})\)	\(q+8.43597i q^{3} +12.2748i q^{5} +1.63924 q^{7} -44.1656 q^{9} +25.7416i q^{11} +13.2187i q^{13} -103.550 q^{15} -53.6113 q^{17} +100.391i q^{19} +13.8286i q^{21} +25.1189 q^{23} -25.6706 q^{25} -144.809i q^{27} -256.105i q^{29} -132.684 q^{31} -217.155 q^{33} +20.1214i q^{35} +247.306i q^{37} -111.512 q^{39} -198.660 q^{41} -404.064i q^{43} -542.124i q^{45} +78.3629 q^{47} -340.313 q^{49} -452.264i q^{51} +743.559i q^{53} -315.973 q^{55} -846.894 q^{57} -65.8036i q^{59} +273.392i q^{61} -72.3982 q^{63} -162.256 q^{65} -399.066i q^{67} +211.903i q^{69} +727.536 q^{71} -106.065 q^{73} -216.556i q^{75} +42.1968i q^{77} -58.9970 q^{79} +29.1298 q^{81} -580.049i q^{83} -658.068i q^{85} +2160.50 q^{87} +768.959 q^{89} +21.6686i q^{91} -1119.32i q^{93} -1232.28 q^{95} -809.953 q^{97} -1136.89i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 28 q^{7} - 54 q^{9}+O(q^{10}) \) 10 * q + 28 * q^7 - 54 * q^9	\( 10 q + 28 q^{7} - 54 q^{9} - 124 q^{15} + 4 q^{17} + 276 q^{23} - 50 q^{25} - 368 q^{31} - 4 q^{33} + 732 q^{39} - 944 q^{47} - 94 q^{49} + 1380 q^{55} - 108 q^{57} - 2628 q^{63} - 492 q^{65} + 3468 q^{71} + 296 q^{73} - 4416 q^{79} - 482 q^{81} + 6036 q^{87} - 88 q^{89} - 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 28 * q^7 - 54 * q^9 - 124 * q^15 + 4 * q^17 + 276 * q^23 - 50 * q^25 - 368 * q^31 - 4 * q^33 + 732 * q^39 - 944 * q^47 - 94 * q^49 + 1380 * q^55 - 108 * q^57 - 2628 * q^63 - 492 * q^65 + 3468 * q^71 + 296 * q^73 - 4416 * q^79 - 482 * q^81 + 6036 * q^87 - 88 * q^89 - 6900 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more